4
votes
Reference for anti-commutative Binomial Theorem
In section 3 of Scurlock 2020, expressions are derived for ${{n}\choose{k}}_{-1}$, which relate to this question through the definition
$$
(x+y)^{n}=\sum_{k=0}^{n}{{n}\choose{k}}_{-1}x^ky^{n-k}
$$
...
- 40.2k
4
votes
What does "because" mean, in the context of an answer to a mathematical problem?
A true story (just for fun).
Once upon a long time ago, the logician Geoffrey Hunter (the author of that excellent old book Metalogic) told me that he used to give a low-level logic course. Near the ...
- 53.1k
4
votes
Reference or proof for number of permutations of [2n] with longest increasing subsequence of length n.
This can be worked out pretty simply using the Robinson–Schensted correspondence. One of the consequences of this correspondence is that the number of permutations on $m$ letters with longest ...
- 15.9k
3
votes
Accepted
A reference request for $SL(2,q)$ being quasisimple for prime powers $q\ge 4$.
Here are two possibilities.
In Huppert's German book "Endliche Gruppen I", it is proved in Satz (= Theorem) 6.10 on page 181 that ${\rm SL}(n,K)$ is perfect for any field $K$ and $n \ge 2$, ...
- 86.7k
3
votes
Accepted
What does "because" mean, in the context of an answer to a mathematical problem?
I agree with univalence's comment: "$X$ because $Y$" means, typically, and at least in your examples, that $Y$ is a sketch of a proof of $X$, or perhaps the key step or key theorem that ...
- 407k
3
votes
Accepted
Can an equilibrium be unstable and asymptotically stable at the same time?
No, it cannot. By definition, an equilibrium point is asymptotically stable if it is both stable and attractive. Therefore, if the equilibrium point is unstable, it cannot be asymptotically stable.
...
- 6,328
2
votes
Imre Ruzsa Generalisation of Kneser's theorem proof
A proof due to DeVos is found here http://math.colgate.edu/~integers/q7/q7.pdf
- 113
2
votes
Accepted
Uniqueness of interpolation for distinct positive real numbers by non-negative coefficients $x_i$ and $\sum_{i=1}^n x_i =1$
Write $A = \sum_{i=1}^n a_i x_i$. This expression writes the number $A$ as a convex combination of the distinct positive reals $\{ a_i \}$. The question is equivalent to asking: given that there is at ...
- 407k
2
votes
Are there any concrete application of the Lyapunov theorem for LTI systems?
Yes, there are many applications. For instance,
it allows you to derive tests to establish the stability of a system for which the matrix $A$ is uncertain (robust analysis). This can be used to ...
- 6,328
2
votes
Accepted
Does the ultra-weak topology coincide with the weak topology on the unit ball?
This is never true if $A$ is infinite-dimensional. In this case, $A$ is a proper (closed) subspace of $A^{\ast\ast}$. Let $q\colon A^{\ast\ast}\to A^{\ast\ast}/A$ be the quotient map and $\psi\in (A^{\...
- 14.6k
2
votes
Accepted
A Question about the _Mathematics Student Journal_
"Mathematics Student Journal" was a scholarly journal published in United States focused on Education (ISSN is 0095-7089). It began in 1954 and ceased publication in 1973. It was continued ...
- 145k
2
votes
Do there exist mathematical transforms other than the Fourier Transform for which there exists some sort of a fast convolution theorem?
Yes, for the Mellin convolution
$$[f(x) * g(x)](y)=\int\limits_0^\infty f(x)\, g\left(\frac{y}{x}\right)\,\frac{dx}{x}\tag{1}$$
one has
$$\mathcal{M}_y[[f(x) * g(x)](y)](s)=\mathcal{M}_x[f(x)](s)\cdot ...
- 6,653
2
votes
Do there exist mathematical transforms other than the Fourier Transform for which there exists some sort of a fast convolution theorem?
The discrete Fourier transform, say in the form of the evaluation/interpolation ring isomorphism
$$
\mathbb{C}[x]/(x^n-1)\cong\prod_i\mathbb{C}[x]/(x-\zeta^i)\cong\mathbb{C}^n
$$
takes cyclic ...
- 9,559
1
vote
Accepted
Transcendence of periods of the Weierstrass elliptic function
You can find the proof in Siegel, C.L., "Über die Perioden elliptischer Funktionen", Journal für die reine und angewandte Mathematik 167 (1932): 62-69. http://eudml.org/doc/149791.
- 4,361
1
vote
On exponential stability of fixed points
The requirement for exponential stability is a little less strict as your first inequality. Namely, one can also include a constant positive but finite factor $\beta$, such that initial transients are ...
- 11.4k
1
vote
total variation distance: $\max_{A \subseteq \mathcal{A}} \left| P(A)-Q(A) \right| = \frac{1}{2} \sum_{x \in \mathcal{A}} \left| P(x) - Q(x)\right|$?
I would read ``Markov Chains and Mixing Times'', second edition, by David A. Levin, Yuval Peres with contributions by Elizabeth L. Wilmer, Chapter 4. It is available online.
- 11
1
vote
Accepted
When can we say that $a \ge b \ge c$ without loss of generality?
If you have a problem in which three numbers labelled "$a$", "$b$" and "$c$" appear and you know nothing else about them then you know they must be in some linear order. ...
- 89.4k
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